\section{Groups}
\begin{Definition}[Exactness]
Call a sequence 
$$\cdots G_{i-1}\stackrel{\alpha_{i-1}}\longrightarrow G_i\stackrel{\alpha_i}\longrightarrow G_{i+1} \cdots$$
of group homomorphisms exact if $Im(\alpha_{i-1})=Ker(\alpha_i)$.

We call a sequence 
$$G_1\stackrel{\alpha_1}\hookrightarrow G_2 \stackrel{\alpha_2}\twoheadrightarrow G_3$$
a short exact sequence if it is an exact sequence and $\alpha_1$ is injecte and $\alpha_2$ is surjective.
\end{Definition}

\begin{Definition}[Split exact sequence]
We call a short exact sequence 
$$G_1\stackrel{\mu}\hookrightarrow G_2 \stackrel{\epsilon}\twoheadrightarrow G_3$$ 
split if there exists a homomorphism $\sigma:G_3\rightarrow G_2$ with $\epsilon \circ \sigma =id_{G_3}$. We say $\sigma$ splits $\epsilon$.
$\sigma$ embeds $G_3$ in $G_2$ as a subgroup, but not necessarily normal.

In this case $G_2$ is the semidirect product of $G_1$ and $G_3$.

\end{Definition}

\begin{Theorem}[Equivalence of splits]
Let $M'\stackrel{\mu}\hookrightarrow M \stackrel{\epsilon}\twoheadrightarrow M''$ be a short exact sequence of abelian groups.
The following are equivalent:
\begin{enumerate}[i)]
\item The sequence is split at $\mu$.
\item The sequence is split at $\epsilon$.
\item There are homomorphisms $\sigma : M'' \rightarrow M$ and $\pi: M\rightarrow M'$ with $\epsilon\sigma=id_{M''}, \pi\mu=id_{M'}$ and $\mu\pi + \sigma\epsilon=id_M$, that is $\{\mu,\pi,\sigma,\epsilon\}$ is a full set of inclusions and projections on $M$.
\item $M$ is isomorphic to $M'\oplus M''$ through the maps $\mu$ and $\epsilon$, in other words: There are isomorphisms $\phi: M\rightarrow M'\oplus M''$ such that $\phi\mu(m')=(m',0),$ $\epsilon\phi^{-1}(0,m'')=m''$ whenever $m'\in M', m'' \in M''$.
\end{enumerate}
\end{Theorem}

\subsection{Free Groups}
The univsersal property is the most important property of a free group and it is sufficient to characterize free groups.

\begin{Definition}[Free Groups]
Given a set $X$, the free group $Fr(X)$ on $X$ is the uniquely determined (up to isomorphism, proof!) group (together with an inclusion $X \hookrightarrow Fr(X)$) fulfilling the universal property for $X$. That is for any group $G$, the inclusion $X\hookrightarrow Fr(X)$ provides a bijection:
$$ \{set functions X\rightarrow G\} \leftrightarrow \{group homomorphisms Fr(X)\rightarrow G\}$$
$Fr(X)$ is called the free group generated by $X$. A group is called free if it is isomorphic to $Fr(X)$ for some set $X$.
\universal{X}{Fr(X)}{G}{f_x}{\mu}{\exists ! \phi}
\end{Definition}

\begin{Theorem}
Every subgroup of a free group is free.
\end{Theorem}

\begin{Lemma}
In the presentation for $G$ $R \hookrightarrow F\twoheadrightarrow G$ the group $R$ is a free group. Hence often called the free presentation for $G$. 
\end{Lemma}

\begin{Lemma}
Every group is a homomorphic image of a free group.

\end{Lemma}

\begin{Lemma}
Every group has a presentation.
\end{Lemma}
\begin{Lemma}
A group $G$ is free iff every epimorphism $\epsilon: H\twoheadrightarrow G$ splits.
\end{Lemma}


\begin{Theorem}
For an abelian group $A$ with subset $X$ the following are equivalent.
\begin{enumerate}[i)]
\item Every element of $A$ is uniquely expressible as a $\ZZ$-linear combination of elements of $X$.
\item $A$ is isomorphic to the restricted direct product (=direct sum) of infinite cyclic groups.
\item The inclusion $X\hookrightarrow A$ gives rise, for each abelian group $B$ to the diagram:

and so a bijection $$\{set functions X\rightarrow B\}\leftrightarrow\{group homomorphisms A\rightarrow B\}$$
\end{enumerate}
\universal{X}{Fr_{Ab}(X)}{A}{f_x}{\mu}{\exists ! \phi}
In this situation call $A$ the free abelian group generated by $X$ and write $A=Fr_{\ZZ}(x)$.
\end{Theorem}

A group is called simple if it has no nontrivial proper normal subgroup.

\begin{Example}[Simple Groups] Any simple group is one of the following.
\begin{enumerate}
\item Alternating group $A_n,n\geq 5$
\item
For every field $F$ and $n\geq3$ the projective linear group $PSL_n(F)$ is simple. It is obtained from the special linear group $SL_n(F)$ of all determinant one matrices bz factoring out the subgroup of scalar matrices. ($n=2$ works when $|F|\geq3$).
\item Other linear groups (i.e. orthogonal groups, groups of LIE type)
\item 26 sporadic finite simple groups
\item Infinite simple groups
\end{enumerate}
\end{Example}


\begin{Theorem}
Every finite abelian group $G$ uniquely determines and is uniquely determined (up to isomorphism) by its direct sum decomposition.
$$A \cong \bigoplus_{p~prime}(\oplus_{k \geq 1} (\ZZ / p\ZZ)^{r(p)_k})$$
Here $r(p)=\sum_{k\geq 1 r(p)_k}$ is called the $p$-rank of $A$.
\end{Theorem}
\begin{Definition}[metabilian]
A group $G$ is called metabilian if there exists an extension 
$$1\rightarrow abelian \rightarrow G \rightarrow abelian \rightarrow 1$$
\end{Definition}


\begin{Definition}[soluble, solvable]
A group is called soluble/solvable if it can be built by finitely many extensions from abelian groups.
Say it has soluble length $n\geq 1$ if it is nontrivial and $n-1$ extensions suffice and at least $n-1$ extensions are needed. Also say the trivial group is soluble of length 0. Non trivial abelian groups have soluble length 1 and non abelian metabelian groups have soluble length 1.

\end{Definition}

\begin{Lemma}
Let $N\hookrightarrow G \twoheadrightarrow G/N$ be a group extension.
\begin{enumerate}
	\item If $N$ is soluble of length $n$ and $G/N$ is soluble of length $k$ then $G$ is soluble of length $\leq n+k$.
	\item If $G$ is soluble of length $m$, then $N$ and $G/N$ are both soluble with length $\leq m$ 
\end{enumerate}
\end{Lemma}

\begin{Definition}[group series, factors of a series]
Call 
$$1=G_0\triangleleft  G_1 \triangleleft \ldots \triangleleft  G_{n-1}\triangleleft  G_n=G$$
 a series of length $n$ in $G$. 
The factor groups $G_i/G_{i-1}$ are called the factors of the series. 
\begin{enumerate}
	\item normal series: If each $G_{i-1}$ is normal in $G$ 
\item proper series: If all factros are nontrivial
\item composition series: finite length, proper and all factors are simple groups
\end{enumerate}
\end{Definition}

\begin{Lemma}
A group $G$ is soluble iff there is a finite series for $G$ in which every factor is abelian.
\end{Lemma}

\begin{Definition}[commutator, commutator group]
In a group $G$, the commutators are the elements of the form $[g_1,g_2]:=g_1g_2g_1^{-1}g_2^{-1}$.
Since the inverse of a commutator is a commutator, the subgroup generated by all commutators is just the set of all products of commutators.
The subgroup $[G,G]\triangleleft G$ of all commutators is normal in $G$.
$n[g_1,g_2]n^{-1}=[ng_1n^{-1},ng_2n^{-1}]$. $[G,G]$ is a characteristic subgroup, i.e. it is preserved under every automorphism of $G$.
$G/[G,G]$ is abelian.

Iterate: $G^0:=G.  G^n=[G^{n-1},G^{n-1}].$
This gives a descending normal series, which we call the (possibly infinite) derived series.
\end{Definition}

\begin{Lemma}
Suppose $N\triangleleft H$. Then $H/N$ is abelian iff $[H,H]\leq N$
\end{Lemma}

\begin{Lemma}
\begin{enumerate}
	\item $G$ is soluble iff its derived series is finite.
	\item If $G$ is soluble, then its (soluble) length is the length of its derived series.
\end{enumerate}
\end{Lemma}

\begin{Definition}[central extension]
Call an extension $N\hookrightarrow G \twoheadrightarrow G/N$ a central extension when $N$ lies in the center $Z(G):=\{x\in G : \forall g \in G xg=gx\}$.

A group $G$ is called nilpotent if it has a central series of finite length that reaches $G$.
\end{Definition}

\begin{Lemma}
A finite group is nilpotent iff it is isomorphic to a product of groups of prime power order (called $p$-groups).
\end{Lemma}

\begin{Definition}[upper central series, lower central series]
Let $G$ be a group. We denote the following definitions.
\begin{enumerate}
	\item Upper central series: $1=\theta_0(G)\triangleleft \theta_1(G)\triangleleft\ldots$ where $\theta_{i+1}$ is defined by $\theta_{i+1}(G)/\theta_i(G)=Z(G/\theta_i(G))$.
	\item Lower central series: $G=\gamma_1(G)\triangleright\gamma_2(G)\triangleright\ldots$ with $\gamma_{i+1}(G)=[\gamma_i(G),G]$
\end{enumerate}
\end{Definition}

\begin{Lemma}
If $1\triangleleft G_0 \triangleleft G_1\triangleleft \ldots \triangleleft G_n=G$ is a central series in a nilpotent group $G$ then for all $i$
\begin{enumerate}
	\item $G_i\leq \theta_i(G)$ and so $\theta_n(G)=G$
	\item $\gamma(G)\leq G_{n+1-i}$ and so $\gamma_{n+1}(G)=1$.
	\item The upper and lower central series have the same length called the nilpotency class of $G$.
\end{enumerate}
\end{Lemma}

\begin{Definition}[Refinement]
A series $$H: 1=H_0\triangleleft H_1\triangleleft \ldots \triangleleft H_n=G$$ has a refinement
$$ J: 1=J_0\triangleleft J_1\triangleleft \ldots \triangleleft J_m=G$$
if $J$ can be obtained from $H$ by insertion of some extra terms (thus every $H_i$ is some $J_{j_i}$). The refinment $J$ is a proper refinement if it contains a term that is not in $H$.
\end{Definition}

\begin{Lemma}
A proper series (from $1$ to $G$) is a composition series for $G$ iff there is no proper refinement.
\end{Lemma}

\begin{Theorem}[Schreier]
Any two series of a group have equivalent refinements.
\end{Theorem}

\begin{Theorem}
Suppose that $G$ has a composition series $C$.
\begin{enumerate}
	\item Every proper series of $J$ of $G$ has a refinement that is a composition series equivalent to $C$.
	\item (Jordan-Hoelder, 1869, 1889) Any two composition series for $G$ are equivalent.
\end{enumerate}
\end{Theorem}

\subsection{Group Representations}
A representation of a group $G$ is a way of representing $G$ as a group of invertible transformations of some set (possibly with additional structure).

\begin{Example}[basic group actions, set without structure, stabilizer] 
If the set has no extra structure, then its invertible transformations are called permutations and the represenatation is called a permutation representation.

Here let the symmetric group on a set $X$ be the group $Symm(X)$ of all bijections $X\rightarrow X$. Then group actions
$$\alpha: G\times X \rightarrow X, (g,x)\mapsto g\cdot x$$
such that $(1_G,x)\mapsto x$ and $(g_1g_2,x)\mapsto g_1 \cdot (g_2\cdot x)$ corresponding bijectively to group homomorphisms $\rho: G\rightarrow Symm(X)$ via $\alpha(g,x)=\rho(g)(x)$.
We call 
\begin{enumerate}
	\item $X$ a (left) $G$-set and for $x\in X$ the set $Gx=\{gx : g\in G\}$ the orbit of $x$.
	\item $Stab_G(x)=\{g\in G : gx=x\}\leq G$ the stabilizer of $x$.
\end{enumerate}
When $\rho$ is the trivial map, the action is called trivial.

\end{Example}

\begin{Lemma}
Let $x\in X$. Then the function $gStab_G(x)\mapsto g\cdot x$ defines a bijection between the set of left cosets of $Stab_G(x)$ in $G$ and the orbit $Gx$ of $x$.
When $G$ is finite we have $|Gx|=[G: Stab_G(x)]$.
\end{Lemma}

\begin{Example}[Action on a vector space]
When $X$ is a vector space $V$ over a field $K$ and the transformations are linear maps, then the represenation is called a linear representation or $K$-representation and $V$ is called a $G$-space.

Here linear actions $\alpha: G\times V \rightarrow V$ are linear with regard to $K$.
$\rho: G \rightarrow Aut(V)$ via $\alpha(g,x)=\rho(g)(x)$. When $V$ is finite dimensional that is isomorphic to $K^n$ and we can identify $Aut(V)$ with $GL(n,K)$.

In that case $\rho : G \rightarrow GL(n,K)$ is called a matrix representation of dimension or degree $n$.
\end{Example}
\begin{Definition}[group action on $2^G$ by conjugation]
Let $G$ act on $Pot(G)=2^G$ by conjugation. For any $H\subset G$ we call
$Stab_G(H)=\{g\in G : gHg^{-1}=H\}=N_G(H)$ the normalizer of $H$.
\end{Definition}
\begin{Lemma}
Let $G$ be a finite group. For all subsets $H\subset G$, $N_G(H)$ is a subgroup of $G$ and the number of conjugates of $H$ in $G$ equals $[G: N_G(H)]$ and so divides $|G|$.
\end{Lemma}

\begin{Lemma}
$|Gx|=[G: Stab_G(x)]$
\end{Lemma}

\begin{Lemma}
$H\subset G$. The number of conjugates of $H$ in $G$ is $[G:N_G(H)]$ and divides $|G|$.
\end{Lemma}

\begin{Lemma}
If $H,Q$ are subgroups of $G$ with $Q\leq N_G(H)$ then $HQ$ is a subgroup of $G$ with $H\triangleleft HQ$.

EG:  If $H\cap Q=1$ then $HQ=H\ltimes Q$.
\end{Lemma}


\begin{Theorem}[Sylow 1872]
Let $G$ be a group of order $p^rm$ with $p$ prime, $r \geq 1$ and $p \nmid m$. Each subgroup of $G$ of order $p^r$ is called a Sylow $p$-subgroup.
\begin{enumerate}
	\item $G$ has a Sylow $p$-subgroup $P$.
	\item Every $p$-subgroup $Q$ of $G$ lies in a conjugate of $P$ and in particular every Sylow $p$-subgroup is a conjugate to $P$.
	\item The number $n_p$ of Sylow $p$-subgroups is $|Syl_p(G)|\equiv 1 \mod p$
\end{enumerate}

\end{Theorem}


\begin{Lemma}
A finite group is nilpotent iff every Sylow subgroup of $G$ is normal in $G$.
\end{Lemma}

\begin{Lemma}
The group $G$ acts effectively on $X$ iff the corresponding homorphism $\rho: G \rightarrow Symm(X)$ is a monomorphism. In this case the representation $\rho: G\rightarrow Symm(X)$ is called faithful.
\end{Lemma}

\begin{Theorem}[Cayley]
Every group has a faithful permutativ representation. In particular every finite group embeds into some finite symmetric group.
\end{Theorem}

\subsubsection{Complex linear representation of finite groups}
Call representations $\rho_i: G\rightarrow Aut(V_i)$, $i=1,2$ equivalent if there exists a linear isomorphism $\gamma: V_1\rightarrow V_2$ with 
$$\rho_2(g)=\gamma \rho_1(g)\gamma^{-1} ~\forall g\in G$$

Invariants of equivalence classes of linear representations are supposed to stand in bijection to invariants of conjugancy classes (=similarity classes) of matrices.

$tr : M_n(K)\rightarrow K$ is a $K$-linear function with $tr(PQ)=tr(QP)$ and $tr(I_n)=n$. $tr(PAP^{-1})=tr(A)$.
Call $tr \circ \rho : G \rightarrow K$ the character of the representation $\rho$.

\begin{Theorem}
Two complex representations of a finite group are equivalent iff they have the same character.
\end{Theorem}

\begin{Definition}[reducible representation]
A $k$-dimensional subspace $U$ of an $n$-dimensional $G$-space $V$ is called $G$-invariant if $\forall g\in G, u\in U: g\cdot u \in U$.
In this event, when $U\neq 0, U\neq V$, we call the representation on $V$ reducible. When no such (nonzero, proper, invariant) subspace $U$ exists, we call $V$ irreducible $G$-space (with regard to the $G$-action).

Any $1$-dimensional representation is irreducible. Conversely a trivial representation (corresponding to $I_n$) is irreducible iff it's $1$-dimensional.
\end{Definition}

We say the representation on $V$ decomposes as the direct sum of its subrepresentations on $V$ and quotient representation on comlementary subspace to $U$ in $V$. When no such $U$ giving a direct sum decomposition exists, call the $G$-space indecomposable.

When $V$ decomposes into a direct sum of irreducible invariant subspaces, then say the representation is completely reducible.

\begin{Theorem}
Every complex representation of a finite group is completely reducible.
\end{Theorem}
In genereal the standard representation of $S_n$ on $\CCC^n$ decomposes as:
$\CCC^n=span(e_1+\ldots + e_n) \oplus span(e_1-e_n,e_2-e_n,\ldots,e_{n-1}-e_n)$. The first is a $1$-dimensional invariant subspace (trivial representation and the second gives an irreducible representation.
When the order of $|G|=n$, so that $Symm(G)\cong S_n$ and we have the left regular represenataion $G\rightarrow S_3\rightarrow GL_n(\CCC)$ then the second representation becomes decomposable as a $G$-representation.

\begin{Theorem}

\begin{itemize}
	\item When $|G|=n$, the left regular representation of $G$ on $\CCC^n$ decomposes into a direct sum containing every irreducible representation of $G$ (with multiplicity equal to its dimension).
	\item 
	\begin{itemize}
		\item Hence if the irreducible representation of $G$ have dimensions $n_1,\ldots,n_r$ then $n=n_1^2+\ldots+n_r^2$ where $n_1=1$ is the dimension of the trivial representation.
		\item Here, $r$ is the number of conjugancy classes of elements of $G$.
	\end{itemize}
	
\end{itemize}

\end{Theorem}

\begin{Theorem}
Every complex representation of a finite abelian group is a driect sum of $1$-dimensional representations.
\end{Theorem}

\subsection{Monoids}

\begin{Definition}[Monoid]
A Monoid is pretty much a group without inverses, i.e. closed under binary operation, that's associative with identiy element.
\end{Definition}

\begin{Example}

\begin{itemize}
	\item Nonnegative integers under +, identity 0.
	\item $\ZZ$ under multiplication, identity 1
	\item Set of strings on some alphabet, under concatenation. Identity=empty string
	\item Set on $n\times n$ matrices $M_n(K)$ under multiplication, indentity $I_n$.
	\item Set of selfmaps on any mathematical opbejct, under composition. Identity is the identity map.
\end{itemize}

\end{Example}

\begin{Definition}[Monoid homomorphism]
Let $N_1, N_2$ be monoids. 
$\phi : N_1 \rightarrow N_2$ is a monoid homomorphism if
$\forall a,b\in N_1: \phi(a\cdot b)=\phi(a)\cdot \phi(b)$ and $\phi(1_{N_1})=1_{N_2}$.
\end{Definition}

\begin{Definition}[Free monoid]
Suppose given a set $X$. Then the free monoid $F$ on $X$ has the universal property, that every set function $f: X \rightarrow N$ with $N$ a monoid extends to a unique monoid homomorphism $\phi : F \rightarrow N$.

This tells us for all monoids $N$, inclusions $X\hookrightarrow F$ induces a bijection:
$$\{set functions X\rightarrow N\} \leftrightarrow \{monoid homomorphisms F\rightarrow N\}$$
\end{Definition}
Construct the free monoid on $X$ by altering the construction of the free group $Fr(X)$: consider words only on the alphabet $X$ (instead of $X \cup X^{-1}$ in the construction of a free group on $X$)

\begin{Example}
$(\NN,0,+)$ is the free monoid on the singleton set $\{1\}$.
\end{Example}
Books (Algebra): Hungerford, Kostrikin
Books ( Group Theory): Robinson, M. Hall